Background, Uncertainty andImprecision, Statistics and Random Processes, Uncertainty in Information,Fuzzy Sets and Membership, Chance versus Ambiguity.Classical Sets - Operations on Classical Sets, Properties of Classical (Crisp)Sets, Mapping of Classical Sets to FunctionsFuzzy Sets - Fuzzy Set operations, Properties of Fuzzy Sets. Sets as Points inHypercubes
Cartesian Product, Crisp Relations- Cardinality of Crisp Relations, Operations on Crisp Relations, Properties ofCrisp Relations, Composition. Fuzzy Relations - Cardinality of FuzzyRelations, Operations on Fuzzy Relations, Properties of Fuzzy Relations,Fuzzy Cartesian Product and Composition, Non-interactive Fuzzy Sets.Tolerance and Equivalence Relations - Crisp Equivalence Relation, CrispTolerance Relation, Fuzzy Tolerance and Equivalence Relations. ValueAssignments - Cosine Amplitude, Max-min Method, Other Similaritymethods
Features of the Membership Function, StandardForms and Boundaries, Fuzzification, Membership Value Assignments –Intuition, Inference, Rank Ordering, Angular Fuzzy Sets, Neural Networks,Genetic Algorithms, Inductive Reasoning.
Lambda-Cuts for FuzzySets, Lambda-Cuts for Fuzzy Relations, Defuzzification MethodsExtension Principle - Crisp Functions, Mapping and Relations, Functions offuzzy Sets – Extension Principle, Fuzzy Transform (Mapping), PracticalConsiderations, Fuzzy NumbersInterval Analysis in Arithmetic, Approximate Methods of Extension - Vertexmethod, DSW Algorithm, Restricted DSW Algorithm, Comparisons, FuzzyVectors
Classical Predicate Logic – Tautologies,Contradictions, Equivalence, Exclusive OR and Exclusive NOR, LogicalProofs, Deductive Inferences. Fuzzy Logic, Approximate Reasoning, FuzzyTautologies, Contradictions, Equivalence and Logical Proofs, Other forms ofthe Implication Operation, Other forms of the Composition Operation
Natural Language, Linguistic Hedges, Rule-Based Systems - Canonical Rule Forms, Decomposition of Compound Rules,Likelihood and Truth Qualification, Aggregation of Fuzzy Rules, GraphicalTechniques of Inference
Fuzzy Synthetic Evaluation, Fuzzy Ordering,Preference and consensus, Multiobjective Decision Making, Fuzzy BayesianDecision Method, Decision Making under Fuzzy States and Fuzzy Actions.
Classification by Equivalence Relations - CrispRelations, Fuzzy Relations. Cluster Analysis, Cluster Validity, c-MeansClustering - Hard c-Means (HCM), Fuzzy c-Means (FCM). ClassificationMetric, Hardening the Fuzzy c-Partition, Similarity Relations from Clustering